What is an inclusion map? (manifolds)

[Fellowroot]

In my book its says let i: U →M (but with a curved arrow) and calls it an inclusion map. What exactly is an inclusion map? Doesn't the curve arrow mean its 1-1? So are inclusion maps always 1-1?

 

[lavinia]

In my book its says let i: U →M (but with a curved arrow) and calls it an inclusion map. What exactly is an inclusion map? Doesn't the curve arrow mean its 1-1? So are inclusion maps always 1-1?

yes

 

[Fredrik]

The map $I:X\to X$ defined by $I(x)=x$ for all $x\in X$ is called the identity map on $X$. If X is a subset of Y, then the map $J:X\to Y$ defined by $J(x)=x$ for all $x\in X$ is called the inclusion map from $X$ to $Y$. If X=Y, the identity map and the inclusion map are the same. If you're using a definition of "map" such that the codomain isn't one of the things that identify the map, then the identity map and the inclusion map are the same, even when X is a proper subset of Y.

 

[Geometry_dude]

The curved arrow is usually reserved for inclusions. In general, if you have a differentiable manifold $M$ and a subset $N \subseteq M$ that is also a differentiable manifold then the inclusion map
$$\iota \colon N \to M \colon p \to p$$
is open (trivially, with respect to the subspace topology on the image), injective and an immersion (i.e. the differential $\iota_*$ has trivial kernel at every point). The tuple $(N, \iota)$ is an embedded submanifold of $M$ and one can always think of embedded submanifolds as being given by subsets by looking at the image and the inclusion map.

 

[micromass]

The curved arrow is usually reserved for inclusions. In general, if you have a differentiable manifold $M$ and a subset $N \subseteq M$ that is also a differentiable manifold then the inclusion map
$$\iota \colon N \to M \colon p \to p$$
is open

Why would it be open?

 

[Geometry_dude]

It is open with respect to the subspace topology on the image, which just happens to be the topology on the set itself by definition. So actually the map
$$\iota \colon N \to M$$
is only open if $\dim M = \dim N$.

 

[lavinia]

A subset of a set can be viewed as a set in itself. The inclusion map takes its points as a set and maps them to the corresponding points in the subset.

For instance the inclusion of {a} into {a,b} maps {a} to the subset {a} ⊂ {a,b}.

The idea of inclusion applies to sets and subsets. It is completely general and is not restricted to manifolds or topological spaces.